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https://huji.zoom.us/j/82527130931?pwd=tadaUFs1wfrT4WGJHRGEzLtOoFlgo9.1 Title:  Around first-order rigidity of Coxeter groups
Abstract:  We present recent results surrounding the problem of first-order rigidity of finitely generated Coxeter groups. This is the outcome of two joint papers, one with R. Sklinos and one with S. André. The main result of the first paper is a proof of first-order rigidity of affine Coxeter groups, which, by classical results of Oger, lead to a proof of profinite rigidity of affine Coxeter groups, a problem posed by Varghese et al. in recent years. The main result of the second paper is that if (W,S) is a Coxeter system whose irreducible components are either spherical, or affine or (Gromov) hyperbolic, and G is finitely torsion-generated and elementarily equivalent to W, then G is itself a Coxeter group and that furthermore, if W is hyperbolic and even, then W is the only model of its theory to be generated by finitely many torsion elements. This last form of rigidity is the best possible in this context, as by the work of Sela, for any free group F, the universal Coxeter group on three generators W_3 = Z_2 * Z_2 * Z_2 is elementarily equivalent to W_3 * F, and so even Coxeter groups are not first-order rigid in the traditional sense.